Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,18)(11,13,17,15)(12,16,21,24)(19,20,23,22)(25,26)(27,36)(28,33)(29,37)(30,38)(31,34)(32,35)(39,40);(2,5)(6,7)(9,11,14,17)(10,13,18,15)(12,19,21,23)(16,20,24,22)(25,27,32,34)(26,31,35,36)(28,37,39,30)(29,33,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,32,39)(26,33,35,40)(27,37,34,30)(29,31,38,36);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,19)(14,17)(15,18)(16,20)(21,23)(22,24)(25,29)(26,30)(27,33)(28,31)(32,38)(34,40)(35,37)(36,39);(9,12,14,21)(10,16,18,24)(11,19,17,23)(13,20,15,22)(25,30)(26,29)(27,28)(31,33)(32,37)(34,39)(35,38)(36,40);(9,13)(10,17)(11,18)(12,20)(14,15)(16,23)(19,24)(21,22)(25,31)(26,34)(27,35)(28,38)(29,39)(30,33)(32,36)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8344089);